A typical automotive engine creates torque and uses it to spin the crankshaft. This torque is created when a force is applied for a distance. The combustion of gas in a cylinder creates pressure against the piston, which, in turn, creates a force on the piston to push it in a linear motion. The force is transmitted from the piston to a connecting rod, and from the connecting rod into a crankshaft. The connecting rod attaches to the crankshaft some distance from the center of the shaft. The horizontal distance changes as the crankshaft spins, so the torque also changes, since torque equals force multiplied by distance (only in the horizontal component). Only the horizontal distance is important in determining the torque in this type of engine, since when the piston is at the top of its stroke the connecting rod is oriented straight at the center of the crankshaft, and no torque is generated in this position since only the force that acts on the lever in a direction perpendicular to the lever generates torque, and the distance in this point is zero.
Another interesting fact about this type of commonly used linear to rotary converter mechanism is that the displacement of the piston is equal to the diameter of the circular movement, but the torque captured by this type of converter is calculated as a maximum of the radius of this circular movement multiplied by the force applied thereto. This identifies a 50% inefficiency in such systems.
In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. It is comparable to a cycloid but instead of the circle rolling along a line, it rolls within a circle. If the smaller circle has radius r, and the larger circle has radius R=kr, then the parametric equations for the curve can be given by:
            x      ⁡              (        θ        )              =                  r        ⁡                  (                      k            -            1                    )                    ⁢              (                              cos            ⁢                                                  ⁢            θ                    +                                    cos              ⁡                              (                                                      (                                          k                      -                      1                                        )                                    ⁢                  θ                                )                                                    k              -              1                                      )              ,          ⁢            y      ⁡              (        θ        )              =                  r        ⁡                  (                      k            -            1                    )                    ⁢                        (                                    sin              ⁢                                                          ⁢              θ                        -                                          sin                ⁡                                  (                                                            (                                              k                        -                        1                                            )                                        ⁢                    θ                                    )                                                            k                -                1                                              )                .            
If k is an integer, then the curve is closed, and has k cusps—that is, sharp corners where the curve is not differentiable. In the case where a/b=2, the trace of the moving point is a straight line. The arc length and area are therefore given by:
            s      x        =                  8        ⁢                  b          ⁡                      (                          n              -              1                        )                              =                        8          ⁢                      a            ⁡                          (                              n                -                1                            )                                      n                        A      x        =                                        (                          n              -              1                        )                    ⁢                      (                          n              -              2                        )                                    n          2                    ⁢      π      ⁢                          ⁢                        a          2                .            
A 2-cusped hypocycloid is a line segment (Steinhaus 1999, p. 145; Kanas 2003), as can be seen by setting a=2b in equations therefore noting that the equations simplify to:x=a sin θy=0.
The straight line is the only curve for which its tangent coincides with the curve itself. When the center of inversion is not on the line, the inverse of the line is a circle.
If we have a rectangular coordinate system, the horizontal x-axis and the vertical y-axis (the ordinate). Given two points A and B, a straight line through these points makes the shortest path between A and B. The angle of the line with a coordinate axis is constant, and also the derivative of a straight line is constant.
A common mechanism used to convert rotational motion to linear motion is a wheel and rail, such as with railed vehicles or any vehicle driven on a substantially flat surface. If the wheel and road are ideally hard, and the road is level, and disregarding friction losses, the effort to move a load W parallel to the road is zero for any load W. A ball or roller bearing can also be considered in this category of mechanisms, where the road has been “rolled up” into a closed circle. In such a system, the load does not move in the direction of the gravity vector, so the velocity ratio is zero while the advantage is infinite, the limit of the product of these being unity (again, disregarding frictional losses).
The lever or block and tackle is one of the most familiar machines, and its family of related machines is widespread and varied. The lever consists of a lever proper and a fulcrum, to which are applied the load W and the effort F. For an ideal lever 1, when the effort F moves a distance x vertically, the load W moves a distance y=−(b/a)x vertically as well, where a is the distance along the lever between the fulcrum and the force applied to the lever, and where b is the distance along the lever between the fulcrum and the load W. The negative sign indicates that the movements are in opposite directions. For the lever to remain in equilibrium, the moments of the forces acting upon it, with respect to any axis, must be zero. Taking the axis through the fulcrum (which eliminates the reaction force R=F+W) results in Fa−Wb=0, or W=(a/b)F.
The principle of virtual work, very often the easiest way to analyze a machine is, is given by the product of the advantage a/b and the velocity ratio b/a, which is indeed unity (in absolute value). If we multiply the two equation we have obtained by considering the movement and the forces separately, we find Wy=−(b/a)(a/b)Fx, or Wy+Fx=0. If we define the product of a force and a displacement in the direction of the force to be the work done by the force, we conclude that the total work done by all of the external forces in a (small) displacement of a such a machine is zero. Herein, this is what we refer to by a “balanced system” in the subsequent explanation. We see both sides of the conversion of movements, rotary to linear and linear back to rotary, using this same principle.
Analysis of Crank and slider or piston and crank (Rotary to Reciprocating Linear Motion Converter)
A famous mechanism is the crank and slider, which converts reciprocating motion into rotary motion. It was patented by James Pickard in 1780, prompting Watt to devise the sun and planet gear as an alternative. Watt's first rotary engines used the parallel motion and the sun and planet gear. We define x in this type of machine as the displacement of the slider from the position of Front Dead Centre (FDC). The angle the crank has rotated from FDC is θ, and the inclination of the connecting rod is φ. a is given by the length of the crank. The relation between the angles is given by: a sin φ=r sin θ, and x=a(1−cos φ)+r(1−cos θ).
The perimeter of a circle is given by Π multiplied by Diameter. Thus, the displacement of the slider necessary to generate one rotation of the crank is equal to 2 times the diameter of the circle. Therefore the distance traveled by a point in the circumference is greater than the distance traveled by a point in the slider. This difference is represented by Π/2=1.57 (approximately).
When applying forces to this system we can observe that in the input of linear motion, force would have to be applied to a distance of a Diameter twice (or reciprocally) to generate a complete circular motion turn of the crank. But at the output, which is rotational motion, torque is calculated by multiplying the same force applied at a point C by the radius of the circumference as well as passing twice through Y=0 which generates Torque=0. As a conclusion we can see that this system produces a 50% loss of force when converting from linear to rotary motion.
What is needed, therefore, is a motion conversion system that maintains a balanced system between input and output distances, applied forces and motions. This can be accomplished by converting a rotary motion input into a hypocycloid motion having only two cusps (straight line curve). In such a needed device, one rotation of a rotary input shaft would generate at least 32 reciprocal linear cycles internally, which would be converted into at least four rotations of a rotary output shaft while maintaining the same torque characteristics between input and output. Such a needed system would be relatively simple and contain relatively few parts. Further, loses due to friction would be minimal. The present invention accomplishes these objectives.